1?n2解2、?n!n?0?x?1?n2n?x?t ??t??2n?0n!?2?nun?1?0
n??unlim? 收敛域（-∞，+∞）
?tn?n21?n2n令S?t???t????tn
n!n?0n?0n!n?1n!t??e???nn?1?n?1?!tnt?e???nt?n?1?1n?1?n?1?!tn
tn
?e??tn?1?n?1?!??1n?2?n?2?!?et?tet?t2et 故S?x??x?e2?1???xx2??，???,??? ?24??
例、利用计算幂级数的和函数，求下列级数的和
n2?n?1 ???1?n?02n?n??1?n2?n?1??1?解：S????1???n?n?1??????????S1?S2 nn?0n?02?2?n?0?2??nnnS2?
11?12??2 3记：S1?x???n?n?1?xn?x2?n?n?1?xn?2 （-1，1）
n?0n?2? ?x2?n?n?1?xn?2n?2?2???x??? ?x2??xn??x2????x?2??1?x??? ??2x2?1?x?3n?1?4S1???? ?2?27n2?n?14222 ∴ ???1? ???n27327n?0230将函数展开成幂函数
1、泰勒级数与麦克劳林级数
设函数f?x?在x0的某邻城内具有任意阶导数，则级数
f?n??x0?f??x0?f???x0?n2????????x?x?fx?x?x?x?x?0000 n!1!2!h?0?f?n??x0??x?x0?n?? ???n!称为f?x?在x?x0点的泰勒级数 特别当x0?0，则级数
f?n??0?nf??0?f???0?2f?n??0?nx?f?0??x?x???x?? ?n!1!2!n!h?0?称为f?x?的麦克劳林级数
2、函数f?x?展开成泰勒级数的条件?x?x0?R?
f?x?能展开成泰勒级数：f?x???an?x?x0?n?0?nfn?x0??x?x0?n ??n!n?0?收敛于f?x??limRn?x??0
n??Rn?x??f?n?1?????n?1?!?xn?1 ?在x0，x之间
3、幂级数展开式的求法
f?n??x0? 方法1、 直接法：计算an? 证明：limRn?x??0
n??n!及f?x??f?x0??f??x0??x?x0??f???x0??x?x0?2?? 2!f?n??x0??x?x0?n?? ?n! 方法 2、 间接法：利用已知的幂级数展开式，通过变量代换四则运算，逐项求导逐项积分待定系数等方法及到函数的展开式。
例 将下例函数展开成?x?x0?的幂函数 ⑴f?x??ln3x x0?2 ⑵f?x??⑶f?x??
?x?2?解⑴f?x??ln3x?ln?3x?6?6??ln6?1? ?2??x?2?? ?ln6?ln?1??
2??1x?3x?221 x0?1 2x?3 x0?1 ⑷f?x??1?x?1?x?2 x0?0
?x?2????2? ?ln6????1?n?1?nn?1n ?ln6????1?n?1?n12nn?x?2?n 0?x?4
解⑵f?x??111???2x?32?x?1??55121??x?1?5
n1?n2 ????1?n?x?1? 5n?05n ???1?n?0?n2n5n?? x?1n?1?33?x? 22
解⑶f?x??111 ???x?1??x?2?x?1x?2n1111?n?x?1?其中??????1???
x?1?2n?0x?12??x?1???2?2?1??2?? ?1?
x?1?1 ，?1?x?3 2n11111?n?x?1????????1??? x?23??x?1?3?x?1?3n?0?3??1??3?? ?2?x?4
?111?n?1n∴ f?x???????1??n?1?n?1??x?1?
x?1x?2n?03??2 ?1?x?3 （如 x0?0 ，f?x??111 ??x1?x21?2x?22111???? 22x?4x?125x?452x?3 f?x?? 如f?x??ln1?x?x ????1?n?1?n?1?2???1?x3???ln1?x3?ln?1?x? ?ln??1?x?????13nx?xn ??1 ?） ，１n?
解⑷f?x??1?x?1?x?2?2??x?1??1?x?2?2?1?x?2?1 1?x 其中
2?1?x?2?n?1n?1?????2??n?n?1????2x?2nx ??????1?x??n?1?n?0?????x?2??n?1?x??x???2n?1?xn
nnnn?0n?0n?0n?0????f?x??2?nx ?1?x?1
例 P291 例 7.28 7.29
例 模拟试题 习题提示
例 设有两条抛物线 y?nx2?横坐标的绝对值为an
⑴求这两条抛物线所围成的平面图形的面积sn ⑵求级数?11和y??n?1?x2?记它们交点的nn?1sn的和 n?1an?1?2y?nx??1n解：? 得an?
1n?n?1??y??n?1?x2?n?1?∵ 所围平面图形对称y轴 sn?2?an01??212??nx??n?1x???dx nn?1?? ?2? ?an0??1212?12??? ???n?n?1??x??dx?n?n?1???n?n?1?3????n?n?1??341
3n?n?1?n?n?1?
sn44?11???????un an3n?n?1?3?nn?1?sn?lim?u1?u2???u2? n??n?1an? ? ?41?4?lim?1??? 3n???n?1?3